Examples

the set of integers under addition, : here, zero is the identity, and the inverse of an element is .

the set of the positive rational numbers under multiplication, : is the identity, while the inverse of an element is .

for every there exists at least one group with n elements,e.g.,

the set of complex numbers {1, -1, i,-i} under multiplication, where i is the principal square root of -1, the basis of the imaginary numbers. This group is isomorphic to under mod addition.

the Klein four group consists of the set of formal symbols with the relations All elements of the Klein four group (except the identity 1) have order 2. The Klein four group is isomorphic to under mod addition.

the set of "moves" on a Rubik's cube, where a move is understood to be a finite sequence of twists: here, the identity move is to do nothing, while the inverse of a move is to do the move in reverse, thereby undoing it.